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Conference Paper: Geometric structures and substructures on uniruled projective manifolds
Title  Geometric structures and substructures on uniruled projective manifolds 

Authors  
Issue Date  2014 
Citation  Trends in Modern Geometry, Tokyo, Japan, 711 July 2014 How to Cite? 
Abstract  With J.M. Hwang the speaker has developed a geometric theory of uniruled projective manifolds (X) modeled on varieties of minimal rational tangents (mathcal{C}_x(X) subset Bbb PT_x(X)), alias VMRTs. Generalizing works of HwangMok, HongMok considered pairs ((X_0;X)) of uniruled projective manifolds, and established a nonequidimensional CartanFubini Extension Principle (2010) in terms of a certain nondegeneracy condition on the second fundamental form for a pair ((mathcal B subsetmathcal A)) consisting of a VMRT (mathcal A ,) and a linear section (mathcal B ,) of (mathcal A). The latter has led to the characterization of standard embeddings (i: G_0/P_0 hookrightarrow G/P) between rational homogeneous manifolds of Picard number 1 by HongMok (2010) in the longroot and nonlinear cases and by HongPark (2011) in the shortroot cases and in the cases of linear subspaces with identifiable exceptions. The argument therein involving parallel transport of VMRTs has also been applied by HongMok (2013) to establish homological rigidity for certain smooth Schubert cycles.
Recently in a joint work with Y. Zhang we have established a stronger rigidity phenomenon for subVMRT structures, where in place of a germ of mapping (f: (X_0;0) o (X;0)) we consider a germ of submanifold ((S;0) subset (X;0)) for a uniruled projective manifold (X) equipped with a minimal rational component (mathcal K). Defining a subVMRT structure by taking intersections (mathcal C_x(X) cap Bbb PT_x(S)) we have obtained sufficient conditions for (S) to extend to a rationally saturated projective subvariety (Z subset X). In the rational homogeneous case the method yields a strengthening of the results of HongMok and HongPark. For instance, if a germ of submanifold ((S;0) subset (X;0)) inherits by intersecting VMRTs with projectivized tangent subspaces a Grassmann structure of rank (ge 2), then (S) in fact extends to a subGrassmannian in its standard embedding. 
Description  Plenary Lecture  Venue: University of Tokyo 
Persistent Identifier  http://hdl.handle.net/10722/269929 
DC Field  Value  Language 

dc.contributor.author  Mok, N   
dc.date.accessioned  20190516T03:39:19Z   
dc.date.available  20190516T03:39:19Z   
dc.date.issued  2014   
dc.identifier.citation  Trends in Modern Geometry, Tokyo, Japan, 711 July 2014   
dc.identifier.uri  http://hdl.handle.net/10722/269929   
dc.description  Plenary Lecture  Venue: University of Tokyo   
dc.description.abstract  With J.M. Hwang the speaker has developed a geometric theory of uniruled projective manifolds (X) modeled on varieties of minimal rational tangents (mathcal{C}_x(X) subset Bbb PT_x(X)), alias VMRTs. Generalizing works of HwangMok, HongMok considered pairs ((X_0;X)) of uniruled projective manifolds, and established a nonequidimensional CartanFubini Extension Principle (2010) in terms of a certain nondegeneracy condition on the second fundamental form for a pair ((mathcal B subsetmathcal A)) consisting of a VMRT (mathcal A ,) and a linear section (mathcal B ,) of (mathcal A). The latter has led to the characterization of standard embeddings (i: G_0/P_0 hookrightarrow G/P) between rational homogeneous manifolds of Picard number 1 by HongMok (2010) in the longroot and nonlinear cases and by HongPark (2011) in the shortroot cases and in the cases of linear subspaces with identifiable exceptions. The argument therein involving parallel transport of VMRTs has also been applied by HongMok (2013) to establish homological rigidity for certain smooth Schubert cycles. Recently in a joint work with Y. Zhang we have established a stronger rigidity phenomenon for subVMRT structures, where in place of a germ of mapping (f: (X_0;0) o (X;0)) we consider a germ of submanifold ((S;0) subset (X;0)) for a uniruled projective manifold (X) equipped with a minimal rational component (mathcal K). Defining a subVMRT structure by taking intersections (mathcal C_x(X) cap Bbb PT_x(S)) we have obtained sufficient conditions for (S) to extend to a rationally saturated projective subvariety (Z subset X). In the rational homogeneous case the method yields a strengthening of the results of HongMok and HongPark. For instance, if a germ of submanifold ((S;0) subset (X;0)) inherits by intersecting VMRTs with projectivized tangent subspaces a Grassmann structure of rank (ge 2), then (S) in fact extends to a subGrassmannian in its standard embedding.   
dc.language  eng   
dc.relation.ispartof  Trends in Modern Geometry   
dc.title  Geometric structures and substructures on uniruled projective manifolds   
dc.type  Conference_Paper   
dc.identifier.email  Mok, N: nmok@hku.hk   
dc.identifier.authority  Mok, N=rp00763   
dc.identifier.hkuros  237394   